The Cauchy-Kowalevski theorem applied for counting connections with a prescribed Ricci tensor

Abstract

How many linear connections are there with a prescribed Ricci tensor? The question is answered in the analytic case by using the Cauchy-Kowalevski theorem.

How many are projectable classical linear connections with a prescribed Ricci tensor

“…If M is a one point manifold, we get similar results to the ones from [8] (in particular, similar to the ones from [2,3]). If M is a one point manifold and r = 0, we get similar results to the ones from [1].…”

“…The skew-symmetric part of the Ricci tensor of a torsion free classical linear connection is exact (and then closed), see [7]. In [8], the authors proved the following result. Theorem 3.6.…”

“…If n − m ≥ 2, it is equivalent to find all (local) solutions of the system (6) satisfying conditions (2) and conditions (8). In particular, if n − m ≥ 2, any analytical tensor field r on Y of type (0, 2) can be realized locally as the Ricci tensor of an analytic projectable classical linear connection with the trace of torsion tensor equal to τ.…”

Counting holomorphic connections with a prescribed Ricci tensor

“…Similar problems have been studied in many papers, e.g. [1,2,3,5,6,8,9]. For example, in [6], the author studied the existence of local solutions ∇ of the equation Ric ∇ = r with unknown real analytical connection ∇ on a real analytical manifold M , where r is a symmetric real analytical tensor field of type (0, 2) on M .…”

On the existence of connections with a prescribed skew-symmetric Ricci tensor

“…In the analytic situation, the inverse problem was studied in many papers, e.g. [1,2,3,5]. For example, in [5], using the Cauchy-Kowalevski theorem, the authors found (locally) all analytic linear connections for a prescribed analytic Ricci tensor.…”